Vizing's conjecture for cographs

We show that if $G$ is a cograph, that is $P_4$-free, then for any graph $H$, $\gamma (G\mathrm{\square }H)\ge \gamma (G)\gamma (H)$. By the characterization of cographs as a finite sequence of unions and joins of $K_1$, this result easily follows from that of Bartsalkin and German. However, the techniques used are new and may be useful to prove other results.